Chapter 7.4, Problem 26WE

To determine

To Prove: that $\frac{PS}{GJ}=\frac{PQ}{GH}$ in the figure shown such that $\overrightarrow{PS}\text{ and }\overrightarrow{GJ}\text{ are angle bisectors}$ .

Explanation of Solution

Given:

Figure is shown as.

  

  $\overrightarrow{PS}\text{ and }\overrightarrow{GJ}\text{ are angle bisectors}$ and $\Delta PQR\sim \Delta GHI$

Calculation:

Since, $\Delta PQR\sim \Delta GHI$ therefore congruent parts of congruent triangle are congruent.

I.e.

  $\angle Q\cong \angle H$ and $\angle P\cong \angle G$

Since, $\angle P\cong \angle G$ therefore.

  $\begin{array}{l}\angle P\cong \angle G\\ \frac{1}{2}\angle P\cong \frac{1}{2}\angle G\mathrm{.......}\left(1\right)\end{array}$

It is given that $\overrightarrow{PS}\text{ and }\overrightarrow{GJ}\text{ are angle bisectors}$ so,

  $\angle QPS=\frac{1}{2}\angle P\text{and }\angle HGJ=\frac{1}{2}\angle G$

From (1).

  $\begin{array}{l}\frac{1}{2}\angle P\cong \frac{1}{2}\angle G\\ \angle QPS=\angle HGJ\end{array}$

In triangles QPS and HGJ.

  $\begin{array}{l}\angle Q=\angle H\therefore \left(\text{Proved above}\right)\\ \angle QPS=\angle HGJ\therefore \left(\text{Proved above}\right)\end{array}$

Therefore, by AA postulate $\Delta QPS\sim \Delta HGJ$

Ratio of sides of two similar triangle is equal.

  $\frac{PS}{GJ}=\frac{PQ}{GH}$

So, it has been proved.